Mesh density functions based on local bandwidth applied to moving mesh methods

TL;DR

This paper introduces bandwidth-based mesh density functions for moving mesh methods, which adaptively refine meshes based on local solution bandwidth, leading to significantly faster convergence in solving PDEs.

Contribution

The paper proposes a novel bandwidth-based mesh density function for moving mesh methods, applicable beyond spectral methods, and demonstrates its effectiveness on various PDEs.

Findings

Bandwidth mesh density functions improve convergence rates.

Solutions are up to ten times faster than uniform meshes.

Bandwidth functions outperform arclength-based meshes by threefold.

Abstract

Moving mesh methods provide an efficient way of solving partial differential equations for which large, localised variations in the solution necessitate locally dense spatial meshes. In one-dimension, meshes are typically specified using the arclength mesh density function. This choice is well-justified for piecewise polynomial interpolants, but it is only justified for spectral methods when model solutions include localised steep gradients. In this paper, one-dimensional mesh density functions are presented which are based on a spatially localised measure of the bandwidth of the approximated model solution. In considering bandwidth, these mesh density functions are well-justified for spectral methods, but are not strictly tied to the error properties of any particular spatial interpolant, and are hence widely applicable. The bandwidth mesh density functions are demonstrated by applying periodic spectral and finite-difference moving mesh methods to a number of model problems in acoustics. These problems include a heterogeneous advection equation, the viscous Burgers' equation, and the Korteweg-de Vries equation. Simulation results demonstrate solution convergence rates that are up to an order of magnitude faster using the bandwidth mesh density functions than uniform meshes, and around three times faster than those using the arclength mesh density function.